用BOUT++模拟边界局域模的形成
夏天阳1,2, 徐学桥2, Ben Dudson3, Bjorn Sjogreen 2, 李建刚1
1Institute
of Plasma Physics, Chinese Academy of Sciences, Hefei, China.
Livermore National Laboratory, Livermore, CA 94550, USA
3University of York, Heslington, York YO10 5DD, United Kindom
2Lawrence
中科院合肥分院 等离子体所
2011年1月9日
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Security, LLC, Lawrence Livermore National Laboratory under Contract DEAC52-07NA27344, and is supported by the China Natural Science Foundation under Contract No.10721505.
1
Outline
1. Background
2. 4-field peeling-ballooning simulation with BOUT++
3. 5-field simulation
4. Linear simulation on EAST
5. Comparison of numerical schemes in BOUT++
6. Summary
2
Background
In H-mode, the localized edge modes (ELMs) is a
dangerous perturbation for large tokamaks, such as
ITER. ELMs are triggered by ideal MHD instabilities.
The type I ELM is successfully explained by ideal
peeling-ballooning (P-B) theory in pedestal, in which
the steep pressure gradients drive ballooning mode
and bootstrap current generates peeling mode.
ELMs in MAST
BOUT++ simulates the Peeling-Ballooning
modes through two fluid framework,
which could study the nonlinear dynamics
of ELMs including extensions beyond
MHD physics.
*
*Xu, X., et al., Nucl. Fusion 51 103040 (2011)
3
Structure of BOUT++
数据结构：2D，3D场量；
2D，3D矢量
格点，几何位形
代数算符
差分算符
偏微分方程组求
解器(SUNDIAL)
物理模型
4
Geometry of BOUT++
All the simulations for these work are based on the
shift circular cross-section toroidal equilibria
(cbm18_dan8) generated by TOQ codes*. The
equilibrium pressure is the same for all cases.
JET-like aspect ratio
Highly unstable to ballooning modes ( γ~0.2ωA)
Widely used by NIMROD, M3D-c1
θ
ψ
Field aligned coordinate applied in BOUT++:
* Miller R.L. and Dam J.W.V. Nucl. Fusion 27 2101 (1987).
5
Outline
1. Background
2. 4-field peeling-ballooning simulation with BOUT++
3. 5-field simulation
4. Linear simulation on EAST
5. Comparison of numerical schemes in BOUT++
6. Summary
6
4-field peeling-Ballooning model with perturbed
parallel velocity
Previous 3-field model
With perturbed parallel velocity
Hall effect term
Perturbed parallel velocity can stabilize the
peeling-ballooning mode
The parallel flow reduces the growth rate by around 20.0% .
The Hall effect increases the growth rate by 9.8%.
With parallel velocity, the Hall effect will increase the growth rate by 19.1%.
Parallel viscosity
•
The parallel viscosity acts as the diffusion term of parallel velocity.
Linear growth rate is independent of
viscosity coefficient. For classic MHD
model, diffv=0.65 (R20//TA) m2/s.
9
Hyper-diffusion
In order to smooth the zigzags of the profiles
of variables to continue the calculation, we
add the hyper-diffusion terms in the
equations. We hope the hyper-diffusion of
pressure and vorticity will not affect the
growth rate and the ELM structure obviously,
but can smooth the profiles effectively.
10
Outline
1. Background
2. 4-field peeling-ballooning simulation with BOUT++
3. 5-field simulation
4. Linear simulation on EAST
5. Comparison of numerical schemes in BOUT++
6. Summary
11
Theoretic Model for 5-Field in BOUT++
We evolve a set of nonlinear evolution equations for perturbations of the ion number
density ni , ion temperature Ti, electron temperature Te, vorticity and magnetic flux ψ:
12
Effects of the diamagnetic drift in ELMs
 Hastie* pointed out the dispersion relationship of the ballooning mode:
 The growth rate of ideal MHD:
 Neglect the electron part:
-w(w-w*) = gI
 Diamagnetic drift plays as the frequency threshold of the perturbation. Only
the fluctuation with w>w*/2 can grow.
* R. J. Hastie, J. J. Ramos, & F. Porcelli, Phys. Plasmas, 10, 4405 (2003).
13
Diamagnetic stabilization on linear growth rate
for spatial constant n0
For Ideal MHD, spatial constant n0 is equivalent to the previous 3-field results because the
density terms do not appear in ideal MHD.
The dispersion relation is written as
γnc is the growth rate in ideal MHD and γnd is the
growth rate with diamagnetic effects.
The diamagnetic drift term provides a
real frequency and stabilizing effect. So
the linear growth rate is proportional to
n0.
The units of n0_ave is 1020
14
Diamagnetic stabilization on linear growth rate
for spatial constant T0
If T0 is a spatial constant, then the cross term in the Laplace equation of vorticity generate
novel effects.
For Ideal MHD, the cross term just plays
as the oscillatory term, so for this case the
linear growth rate should not changed a lot.
With diamagnetic effects, the linear
growth rate is inversely proportional to T0.
The units of Ti0 is keV
15
General cases for both radially varying n0 and T0
For more general situation, both the equilibrium temperature and density profiles are not
constant in radial direction. Here we still apply the same pressure profile, but introduce the
profile for ion density:
There are 5 density profiles are applied in this work:
Table 1. The parameters of five models of different equilibrium
ion density profile.
16
Diamagnetic stabilization on linear growth rate for
both radially varying n0 and T0
The equilibrium density profiles for the
cases in Table 1.
The linear growth rate for the cases
in table 1. For Ideal MHD model, the
linear growth rate of case 0’ is larger
than case 0 by 6.2%. With
diamagnetic effects, the percentage
goes to 31.4% for n=15
17
The cross term in vorticity can enlarge growth rate
For spatial constant T0, the Fourier analysis will give the angular frequency as:
Cross term
Then the linear growth rate is easily obtained as:
This equation shows the qualitatively same results as our simulations
The similar results have
been observed using
MINERVA code*. (a) is for
spatial constant n0 and (b) for
a radially varying n0.
N. Aiba, M. Furukawa, M. Hirota & S. Tokuda, Nucl. Fusion 50 045002 (2010) .
18
Radial pressure profile revolution during ELM crashes
Comparison of the radial pressure profiles on the outer mid-plane for the different cases for n=15.
Case 1, 2 and 3 use Neumann boundary in the core region and 1’, 2’ and 3’ apply Dirichlet boundary condition.
At the early nonlinear phase after ELM crashes, T=95TA, the collapse keeps localized around the peak gradient region
for all the cases.
At t=175TA, the perturbations go into the core boundary except the constant n0 case. This is because the cross term in
the vorticity equation provides an additional drive on the radial direction.
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ELM size evolution after ELM crashes
In order to describe the nonlinear effects, we define the ELM size as
This figure shows that the ELM size is larger for a radially varying n0 than for the spatial
constant density case. This is related to the results on the previous page, which showed that
the cross term in the vorticity equation provides an additional drive on the radial direction. 20
The poloidal distribution of density perturbation
Case 0, without density gradient.
Case 2, with density gradient.
With density gradient, the rotating direction will be different in linear phase.
In nonlinear phase the perturbation will go into the core region.
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Outline
1. Background
2. 4-field peeling-ballooning simulation with BOUT++
3. 5-field simulation
4. Linear simulation on EAST
5. Comparison of numerical schemes in BOUT++
6. Summary
*
22
EAST data for the linear simulations with 3-field
model*
Typical parameters of EAST:
Major radius R0: 1.88m
Minor radius a: 0.43m
Toroidal Field BT: 2.0T
Current Ip: 0.5MA
Safety factor q95: 7.73
Electron temperature Te: 5.11keV
Shot: #33068
The MHD equilibrium of EAST experiment is
calculated by conventional EFIT. The pressure
profile is not consistent with the measurement
from experiment.
*Collaboration with Zixi Liu & Shaocheng Liu of EAST.
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Linear simulations on EAST with 3-field model*
We use the magnetic geometry data in g-file and
experimentally measured pressure and calculated
current from the pressure.
To validate this method, the data of H-mode of C-mod
is tested. We use the H-mode pressure and current to
derive the similar results.
S=108, ELMy free
S<=107, ELMy
Diamagnetic
effect is obvious in
EAST.
*Collaboration with Zixi Liu & Shaocheng Liu.
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Outline
1. Background
2. 4-field peeling-ballooning simulation with BOUT++
3. 5-field simulation
4. Linear simulation on EAST
5. Comparison of numerical schemes in BOUT++
6. Summary
*
25
Comparison of differencing schemes for ELM
simulations
For the purpose of broad application of BOUT++, various differencing schemes has been
implemented. Here we compare the ELM simulations of 3-field model with several
differencing methods to find the most robust scheme in ELM simulations.
The schemes used here are all for the nonlinear Poisson bracket. It means the ExB drift
~
(
) and magnetic flutter b   (
). Because the simple methods of straight
central or upwinding schemes for both terms will result in the code crashes in the early
nonlinear phase, the special schemes are chosen:
 3rd order WENO: Weighted Essentially Non-Oscillatory, a nonlinear adaptive procedure
avoiding crossing discontinuities in the interpolation procedure as much as possible
Arakawa: original version is second order, keep the commutative property of Poisson
bracket, the square vorticity conservation and kinetic energy conservation. In this work, we
implement the 4th order Arakawa in the code for the first time.
CTU: Corner Transport Upwind, 1st order, takes into account the effect of information
propagating across corners of zones in calculating the flux
1st and 4th order Upwind
4th order central
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Linear growth rates for different schemes are
converged to radial resolutions
Each the schemes obtain
convergence.
Convergence for the different
schemes is expected except for the 1st
order upwinding for ExB drift, the
reason is under investigation.
The convergent schemes show the
differences less than 3%.
Linear growth rate vs. radial grid resolution.
The schemes are labeled as X+Y:
X is the scheme for ExB drift.
Y is for magnetic flutter term.
27
Nonlinear:higher
higherorder
orderExB
ExBconvection
convectionschemes
schemes
Nonlinear:
yield saturate pressure
yield saturate pressure
The nonlinear regime shows a
more complicated convergence.
The nonlinear regime shows a
For high resolution, 3rd WENO +
more
complicated
convergence.
4th order
central or
+ upwinding
shows good convergence with 3rd
WENO + Arakawa. Therddifference is
For high
3 WENO
WENO +
less
than resolution,
3.5%. The full
is largerorby
15.0%
4thscheme
order central
+ upwinding
rd
shows
good
convergence
with
3
The schemes for ExB drift
WENO
+ Arakawa.
The difference
determine
the saturate
pressureis
~
more
b   . Lower
less
thaneffectively
3.5%. Thethan
full WENO
order schemes for EXB obtain the
scheme
is larger by 15.0%
lower pressure.
The schemes for ExB drift
determine the saturate pressure
28
Nonlinear: Saturate pressure for different
resolution is grouped by the order of schemes
Poloidal resolution
Radial resolution
29
Nonlinear: ELM size with different resolutions is
also grouped by the order of schemes
Poloidal resolution
Radial resolution
30
Nonlinear: kr power spectra of perturbed pressure are
converged for 5<kr<80
All the schemes show the similar spectrum at medium kr: 5<kr<80.
For both resolutions, the 4th upwinding + Arakawa shows a larger spectrum at high kr.
For the lower resolution at at high kr, these WENO+ Arakawa and WENO+WENO get much larger spectra than the
other schemes.
For the higher resolution case, WENO + Arakawa has the spectrum more close to full WENO. At high kr end, both
spectrums are damped and similar to other schemes.
Here kr is the mode number
31
Nonlinear: kq power spectra of perturbed pressure
shows consistency at high poloidal resolution
All the spectra shows consistency at poloidal direction except the first order schemes and 3rd WENO +
3rd WENO for 512x64x64 resolution.
Increasing the poloidal resolution, all the 3rd orders schemes show consistency.
32
Lower order + Arakawa scheme works well without
hyper-resistivity for low radial resolutions
Lower grid resolution require larger hyper-resistivity:
SH=1011 is necessary for 260x64 grid and SH=1.43x1012 for 1028x64 case, while SH=1x1012 for 516x64.
If we change the scheme for magnetic flutter from WENO to Arakawa, the simulation can work well for the
resolutions below 1028 even without ηH. However, ηH is necessary for 1028x64.
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Summary
4-field:
Perturbed parallel velocity will decrease the linear growth rate by 20.0%
5-field:
n0 will not affect the linear growth rate in Ideal MHD model. With diamagnetic effects,
the growth rate is inversely proportional to n0.
For the same profile of P0 and T0 case, the cross term in vorticity enlarges the growth
rate by 6.2% compared with spatial constant n0 case in Ideal MHD. With diamagnetic
effects, the percentage is 31.4% for n=15.
The density gradient will drive the perturbation into the core region and give a larger
ELM size.
Schemes comparison:
For linear regime, most of the combinations of the schemes we tested in BOUT++ give
the same results.
For nonlinear regime, the order of the schemes for ExB determines the results.
The Arakawa schemes in nonlinear magnetic flutter help the simulations to go
through ELM crashes.
The kr power spectra show that all the schemes are similar at medium kr. The kq
spectra shows that all the 3rd order schemes converge at high poloidal resolution.
34